Having survived an early penalty appeal against him for handball, Alcaraz was forced off after 32 minutes with a shoulder problem at Goodison Park, and Martinez fears that he could be on the sidelines indefinitely.
He told reporters: "Alcaraz is injured and going to be out for a long time. It is a problem with a joint in his shoulder. It is one of those awkward accidents that happens in football and we wish him well."
Martinez added: "I haven't seen the replay [of the alleged handball from Alcaraz]. I'm disappointed by the game as I thought it was going to be more of a spectacle but it didn't turn out to be. I thought we were a little frustrated.
"You want to win the game. You are looking at statistics before the sending off for Swansea and we were in control. Overall, it was a good clean sheet because you can leave yourselves exposed. Maybe our tempo in possession was not as sharp as it could have been."
Alcaraz, who has started just eight Premier League games for the Toffees since signing from Wigan Athletic in 2013, joins England youngster John Stones on Everton's list of defensive injury casualties.